Properties

Label 2.88.a.b
Level 2
Weight 88
Character orbit 2.a
Self dual Yes
Analytic conductor 95.867
Analytic rank 0
Dimension 4
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 88 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(95.8667262922\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{45}\cdot 3^{15}\cdot 5^{4}\cdot 7^{2}\cdot 11\cdot 17\cdot 29 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(-8796093022208 q^{2}\) \(+(\)\(10\!\cdots\!32\)\( - \beta_{1}) q^{3}\) \(+\)\(77\!\cdots\!64\)\( q^{4}\) \(+(-\)\(10\!\cdots\!90\)\( - 1822922575 \beta_{1} + \beta_{2}) q^{5}\) \(+(-\)\(88\!\cdots\!56\)\( + 8796093022208 \beta_{1}) q^{6}\) \(+(\)\(21\!\cdots\!76\)\( + 1846032815820606 \beta_{1} + 714006 \beta_{2} + 4213 \beta_{3}) q^{7}\) \(-\)\(68\!\cdots\!12\)\( q^{8}\) \(+(\)\(20\!\cdots\!37\)\( - 70457522433696719338 \beta_{1} - 19291700786 \beta_{2} + 197927972 \beta_{3}) q^{9}\) \(+O(q^{10})\) \( q\) \(-8796093022208 q^{2}\) \(+(\)\(10\!\cdots\!32\)\( - \beta_{1}) q^{3}\) \(+\)\(77\!\cdots\!64\)\( q^{4}\) \(+(-\)\(10\!\cdots\!90\)\( - 1822922575 \beta_{1} + \beta_{2}) q^{5}\) \(+(-\)\(88\!\cdots\!56\)\( + 8796093022208 \beta_{1}) q^{6}\) \(+(\)\(21\!\cdots\!76\)\( + 1846032815820606 \beta_{1} + 714006 \beta_{2} + 4213 \beta_{3}) q^{7}\) \(-\)\(68\!\cdots\!12\)\( q^{8}\) \(+(\)\(20\!\cdots\!37\)\( - 70457522433696719338 \beta_{1} - 19291700786 \beta_{2} + 197927972 \beta_{3}) q^{9}\) \(+(\)\(88\!\cdots\!20\)\( + \)\(16\!\cdots\!00\)\( \beta_{1} - 8796093022208 \beta_{2}) q^{10}\) \(+(\)\(74\!\cdots\!12\)\( + \)\(61\!\cdots\!05\)\( \beta_{1} - 111812372519476 \beta_{2} + 1078925628602 \beta_{3}) q^{11}\) \(+(\)\(77\!\cdots\!48\)\( - \)\(77\!\cdots\!64\)\( \beta_{1}) q^{12}\) \(+(-\)\(44\!\cdots\!78\)\( + \)\(24\!\cdots\!97\)\( \beta_{1} + 448615718005475761 \beta_{2} + 1937036307534328 \beta_{3}) q^{13}\) \(+(-\)\(18\!\cdots\!08\)\( - \)\(16\!\cdots\!48\)\( \beta_{1} - 6280463194414645248 \beta_{2} - 37057939902562304 \beta_{3}) q^{14}\) \(+(\)\(84\!\cdots\!20\)\( + \)\(14\!\cdots\!50\)\( \beta_{1} + \)\(29\!\cdots\!62\)\( \beta_{2} + 1720528777181671875 \beta_{3}) q^{15}\) \(+\)\(59\!\cdots\!96\)\( q^{16}\) \(+(-\)\(17\!\cdots\!54\)\( + \)\(40\!\cdots\!90\)\( \beta_{1} - \)\(15\!\cdots\!82\)\( \beta_{2} - 82346674465175417836 \beta_{3}) q^{17}\) \(+(-\)\(18\!\cdots\!96\)\( + \)\(61\!\cdots\!04\)\( \beta_{1} + \)\(16\!\cdots\!88\)\( \beta_{2} - \)\(17\!\cdots\!76\)\( \beta_{3}) q^{18}\) \(+(\)\(95\!\cdots\!40\)\( - \)\(10\!\cdots\!13\)\( \beta_{1} + \)\(86\!\cdots\!84\)\( \beta_{2} + \)\(15\!\cdots\!82\)\( \beta_{3}) q^{19}\) \(+(-\)\(78\!\cdots\!60\)\( - \)\(14\!\cdots\!00\)\( \beta_{1} + \)\(77\!\cdots\!64\)\( \beta_{2}) q^{20}\) \(+(-\)\(74\!\cdots\!68\)\( - \)\(49\!\cdots\!56\)\( \beta_{1} + \)\(12\!\cdots\!84\)\( \beta_{2} + \)\(62\!\cdots\!32\)\( \beta_{3}) q^{21}\) \(+(-\)\(65\!\cdots\!96\)\( - \)\(54\!\cdots\!40\)\( \beta_{1} + \)\(98\!\cdots\!08\)\( \beta_{2} - \)\(94\!\cdots\!16\)\( \beta_{3}) q^{22}\) \(+(\)\(66\!\cdots\!32\)\( + \)\(82\!\cdots\!70\)\( \beta_{1} + \)\(43\!\cdots\!74\)\( \beta_{2} + \)\(18\!\cdots\!27\)\( \beta_{3}) q^{23}\) \(+(-\)\(68\!\cdots\!84\)\( + \)\(68\!\cdots\!12\)\( \beta_{1}) q^{24}\) \(+(\)\(67\!\cdots\!75\)\( + \)\(96\!\cdots\!00\)\( \beta_{1} - \)\(19\!\cdots\!60\)\( \beta_{2} + \)\(24\!\cdots\!00\)\( \beta_{3}) q^{25}\) \(+(\)\(38\!\cdots\!24\)\( - \)\(21\!\cdots\!76\)\( \beta_{1} - \)\(39\!\cdots\!88\)\( \beta_{2} - \)\(17\!\cdots\!24\)\( \beta_{3}) q^{26}\) \(+(\)\(24\!\cdots\!00\)\( - \)\(28\!\cdots\!82\)\( \beta_{1} + \)\(40\!\cdots\!24\)\( \beta_{2} - \)\(11\!\cdots\!98\)\( \beta_{3}) q^{27}\) \(+(\)\(16\!\cdots\!64\)\( + \)\(14\!\cdots\!84\)\( \beta_{1} + \)\(55\!\cdots\!84\)\( \beta_{2} + \)\(32\!\cdots\!32\)\( \beta_{3}) q^{28}\) \(+(-\)\(23\!\cdots\!10\)\( - \)\(10\!\cdots\!47\)\( \beta_{1} + \)\(71\!\cdots\!09\)\( \beta_{2} + \)\(82\!\cdots\!32\)\( \beta_{3}) q^{29}\) \(+(-\)\(74\!\cdots\!60\)\( - \)\(12\!\cdots\!00\)\( \beta_{1} - \)\(25\!\cdots\!96\)\( \beta_{2} - \)\(15\!\cdots\!00\)\( \beta_{3}) q^{30}\) \(+(\)\(29\!\cdots\!92\)\( - \)\(52\!\cdots\!36\)\( \beta_{1} - \)\(16\!\cdots\!96\)\( \beta_{2} + \)\(47\!\cdots\!92\)\( \beta_{3}) q^{31}\) \(-\)\(52\!\cdots\!68\)\( q^{32}\) \(+(-\)\(24\!\cdots\!16\)\( - \)\(15\!\cdots\!46\)\( \beta_{1} + \)\(23\!\cdots\!02\)\( \beta_{2} - \)\(26\!\cdots\!04\)\( \beta_{3}) q^{33}\) \(+(\)\(15\!\cdots\!32\)\( - \)\(35\!\cdots\!20\)\( \beta_{1} + \)\(13\!\cdots\!56\)\( \beta_{2} + \)\(72\!\cdots\!88\)\( \beta_{3}) q^{34}\) \(+(\)\(35\!\cdots\!60\)\( - \)\(35\!\cdots\!00\)\( \beta_{1} + \)\(33\!\cdots\!16\)\( \beta_{2} - \)\(40\!\cdots\!00\)\( \beta_{3}) q^{35}\) \(+(\)\(15\!\cdots\!68\)\( - \)\(54\!\cdots\!32\)\( \beta_{1} - \)\(14\!\cdots\!04\)\( \beta_{2} + \)\(15\!\cdots\!08\)\( \beta_{3}) q^{36}\) \(+(\)\(45\!\cdots\!66\)\( - \)\(82\!\cdots\!55\)\( \beta_{1} - \)\(23\!\cdots\!67\)\( \beta_{2} + \)\(33\!\cdots\!84\)\( \beta_{3}) q^{37}\) \(+(-\)\(83\!\cdots\!20\)\( + \)\(88\!\cdots\!04\)\( \beta_{1} - \)\(76\!\cdots\!72\)\( \beta_{2} - \)\(13\!\cdots\!56\)\( \beta_{3}) q^{38}\) \(+(-\)\(12\!\cdots\!96\)\( - \)\(83\!\cdots\!86\)\( \beta_{1} + \)\(66\!\cdots\!50\)\( \beta_{2} + \)\(14\!\cdots\!75\)\( \beta_{3}) q^{39}\) \(+(\)\(68\!\cdots\!80\)\( + \)\(12\!\cdots\!00\)\( \beta_{1} - \)\(68\!\cdots\!12\)\( \beta_{2}) q^{40}\) \(+(\)\(13\!\cdots\!02\)\( + \)\(20\!\cdots\!64\)\( \beta_{1} + \)\(16\!\cdots\!08\)\( \beta_{2} + \)\(30\!\cdots\!84\)\( \beta_{3}) q^{41}\) \(+(\)\(65\!\cdots\!44\)\( + \)\(43\!\cdots\!48\)\( \beta_{1} - \)\(11\!\cdots\!72\)\( \beta_{2} - \)\(54\!\cdots\!56\)\( \beta_{3}) q^{42}\) \(+(-\)\(39\!\cdots\!88\)\( + \)\(71\!\cdots\!13\)\( \beta_{1} + \)\(51\!\cdots\!04\)\( \beta_{2} - \)\(32\!\cdots\!08\)\( \beta_{3}) q^{43}\) \(+(\)\(57\!\cdots\!68\)\( + \)\(47\!\cdots\!20\)\( \beta_{1} - \)\(86\!\cdots\!64\)\( \beta_{2} + \)\(83\!\cdots\!28\)\( \beta_{3}) q^{44}\) \(+(-\)\(34\!\cdots\!30\)\( - \)\(14\!\cdots\!75\)\( \beta_{1} + \)\(21\!\cdots\!57\)\( \beta_{2} + \)\(11\!\cdots\!00\)\( \beta_{3}) q^{45}\) \(+(-\)\(58\!\cdots\!56\)\( - \)\(72\!\cdots\!60\)\( \beta_{1} - \)\(38\!\cdots\!92\)\( \beta_{2} - \)\(16\!\cdots\!16\)\( \beta_{3}) q^{46}\) \(+(-\)\(29\!\cdots\!84\)\( - \)\(51\!\cdots\!00\)\( \beta_{1} - \)\(12\!\cdots\!28\)\( \beta_{2} - \)\(16\!\cdots\!94\)\( \beta_{3}) q^{47}\) \(+(\)\(60\!\cdots\!72\)\( - \)\(59\!\cdots\!96\)\( \beta_{1}) q^{48}\) \(+(\)\(11\!\cdots\!33\)\( - \)\(71\!\cdots\!04\)\( \beta_{1} - \)\(16\!\cdots\!12\)\( \beta_{2} + \)\(19\!\cdots\!24\)\( \beta_{3}) q^{49}\) \(+(-\)\(59\!\cdots\!00\)\( - \)\(84\!\cdots\!00\)\( \beta_{1} + \)\(17\!\cdots\!80\)\( \beta_{2} - \)\(21\!\cdots\!00\)\( \beta_{3}) q^{50}\) \(+(-\)\(22\!\cdots\!28\)\( + \)\(22\!\cdots\!46\)\( \beta_{1} - \)\(12\!\cdots\!56\)\( \beta_{2} - \)\(83\!\cdots\!38\)\( \beta_{3}) q^{51}\) \(+(-\)\(34\!\cdots\!92\)\( + \)\(18\!\cdots\!08\)\( \beta_{1} + \)\(34\!\cdots\!04\)\( \beta_{2} + \)\(14\!\cdots\!92\)\( \beta_{3}) q^{52}\) \(+(-\)\(80\!\cdots\!78\)\( + \)\(96\!\cdots\!21\)\( \beta_{1} - \)\(97\!\cdots\!23\)\( \beta_{2} + \)\(10\!\cdots\!96\)\( \beta_{3}) q^{53}\) \(+(-\)\(21\!\cdots\!00\)\( + \)\(24\!\cdots\!56\)\( \beta_{1} - \)\(35\!\cdots\!92\)\( \beta_{2} + \)\(10\!\cdots\!84\)\( \beta_{3}) q^{54}\) \(+(-\)\(25\!\cdots\!80\)\( - \)\(86\!\cdots\!50\)\( \beta_{1} + \)\(63\!\cdots\!42\)\( \beta_{2} - \)\(10\!\cdots\!75\)\( \beta_{3}) q^{55}\) \(+(-\)\(14\!\cdots\!12\)\( - \)\(12\!\cdots\!72\)\( \beta_{1} - \)\(48\!\cdots\!72\)\( \beta_{2} - \)\(28\!\cdots\!56\)\( \beta_{3}) q^{56}\) \(+(\)\(61\!\cdots\!80\)\( - \)\(16\!\cdots\!42\)\( \beta_{1} + \)\(63\!\cdots\!34\)\( \beta_{2} + \)\(13\!\cdots\!32\)\( \beta_{3}) q^{57}\) \(+(\)\(20\!\cdots\!80\)\( + \)\(90\!\cdots\!76\)\( \beta_{1} - \)\(63\!\cdots\!72\)\( \beta_{2} - \)\(72\!\cdots\!56\)\( \beta_{3}) q^{58}\) \(+(-\)\(20\!\cdots\!20\)\( + \)\(84\!\cdots\!01\)\( \beta_{1} - \)\(20\!\cdots\!12\)\( \beta_{2} - \)\(46\!\cdots\!76\)\( \beta_{3}) q^{59}\) \(+(\)\(65\!\cdots\!80\)\( + \)\(11\!\cdots\!00\)\( \beta_{1} + \)\(22\!\cdots\!68\)\( \beta_{2} + \)\(13\!\cdots\!00\)\( \beta_{3}) q^{60}\) \(+(\)\(22\!\cdots\!82\)\( + \)\(28\!\cdots\!93\)\( \beta_{1} - \)\(58\!\cdots\!67\)\( \beta_{2} - \)\(12\!\cdots\!16\)\( \beta_{3}) q^{61}\) \(+(-\)\(26\!\cdots\!36\)\( + \)\(46\!\cdots\!88\)\( \beta_{1} + \)\(14\!\cdots\!68\)\( \beta_{2} - \)\(42\!\cdots\!36\)\( \beta_{3}) q^{62}\) \(+(\)\(17\!\cdots\!12\)\( + \)\(35\!\cdots\!86\)\( \beta_{1} + \)\(25\!\cdots\!14\)\( \beta_{2} + \)\(13\!\cdots\!97\)\( \beta_{3}) q^{63}\) \(+\)\(46\!\cdots\!44\)\( q^{64}\) \(+(\)\(28\!\cdots\!20\)\( - \)\(16\!\cdots\!00\)\( \beta_{1} - \)\(17\!\cdots\!68\)\( \beta_{2} - \)\(46\!\cdots\!00\)\( \beta_{3}) q^{65}\) \(+(\)\(21\!\cdots\!28\)\( + \)\(13\!\cdots\!68\)\( \beta_{1} - \)\(20\!\cdots\!16\)\( \beta_{2} + \)\(23\!\cdots\!32\)\( \beta_{3}) q^{66}\) \(+(-\)\(16\!\cdots\!24\)\( - \)\(15\!\cdots\!21\)\( \beta_{1} + \)\(75\!\cdots\!72\)\( \beta_{2} + \)\(99\!\cdots\!06\)\( \beta_{3}) q^{67}\) \(+(-\)\(13\!\cdots\!56\)\( + \)\(31\!\cdots\!60\)\( \beta_{1} - \)\(11\!\cdots\!48\)\( \beta_{2} - \)\(63\!\cdots\!04\)\( \beta_{3}) q^{68}\) \(+(-\)\(36\!\cdots\!76\)\( - \)\(82\!\cdots\!76\)\( \beta_{1} + \)\(62\!\cdots\!92\)\( \beta_{2} - \)\(15\!\cdots\!84\)\( \beta_{3}) q^{69}\) \(+(-\)\(31\!\cdots\!80\)\( + \)\(31\!\cdots\!00\)\( \beta_{1} - \)\(29\!\cdots\!28\)\( \beta_{2} + \)\(35\!\cdots\!00\)\( \beta_{3}) q^{70}\) \(+(-\)\(45\!\cdots\!48\)\( - \)\(18\!\cdots\!78\)\( \beta_{1} - \)\(58\!\cdots\!86\)\( \beta_{2} - \)\(74\!\cdots\!03\)\( \beta_{3}) q^{71}\) \(+(-\)\(13\!\cdots\!44\)\( + \)\(47\!\cdots\!56\)\( \beta_{1} + \)\(13\!\cdots\!32\)\( \beta_{2} - \)\(13\!\cdots\!64\)\( \beta_{3}) q^{72}\) \(+(\)\(88\!\cdots\!22\)\( + \)\(21\!\cdots\!66\)\( \beta_{1} - \)\(76\!\cdots\!38\)\( \beta_{2} + \)\(77\!\cdots\!76\)\( \beta_{3}) q^{73}\) \(+(-\)\(40\!\cdots\!28\)\( + \)\(72\!\cdots\!40\)\( \beta_{1} + \)\(20\!\cdots\!36\)\( \beta_{2} - \)\(29\!\cdots\!72\)\( \beta_{3}) q^{74}\) \(+(\)\(17\!\cdots\!00\)\( - \)\(92\!\cdots\!75\)\( \beta_{1} - \)\(34\!\cdots\!20\)\( \beta_{2} - \)\(28\!\cdots\!00\)\( \beta_{3}) q^{75}\) \(+(\)\(73\!\cdots\!60\)\( - \)\(78\!\cdots\!32\)\( \beta_{1} + \)\(66\!\cdots\!76\)\( \beta_{2} + \)\(11\!\cdots\!48\)\( \beta_{3}) q^{76}\) \(+(\)\(98\!\cdots\!12\)\( + \)\(65\!\cdots\!20\)\( \beta_{1} - \)\(11\!\cdots\!52\)\( \beta_{2} + \)\(57\!\cdots\!04\)\( \beta_{3}) q^{77}\) \(+(\)\(11\!\cdots\!68\)\( + \)\(73\!\cdots\!88\)\( \beta_{1} - \)\(58\!\cdots\!00\)\( \beta_{2} - \)\(12\!\cdots\!00\)\( \beta_{3}) q^{78}\) \(+(\)\(39\!\cdots\!80\)\( + \)\(49\!\cdots\!28\)\( \beta_{1} + \)\(21\!\cdots\!88\)\( \beta_{2} - \)\(82\!\cdots\!26\)\( \beta_{3}) q^{79}\) \(+(-\)\(60\!\cdots\!40\)\( - \)\(10\!\cdots\!00\)\( \beta_{1} + \)\(59\!\cdots\!96\)\( \beta_{2}) q^{80}\) \(+(-\)\(49\!\cdots\!19\)\( + \)\(22\!\cdots\!50\)\( \beta_{1} + \)\(16\!\cdots\!02\)\( \beta_{2} - \)\(35\!\cdots\!04\)\( \beta_{3}) q^{81}\) \(+(-\)\(12\!\cdots\!16\)\( - \)\(17\!\cdots\!12\)\( \beta_{1} - \)\(14\!\cdots\!64\)\( \beta_{2} - \)\(26\!\cdots\!72\)\( \beta_{3}) q^{82}\) \(+(\)\(27\!\cdots\!92\)\( + \)\(13\!\cdots\!63\)\( \beta_{1} - \)\(31\!\cdots\!68\)\( \beta_{2} + \)\(15\!\cdots\!36\)\( \beta_{3}) q^{83}\) \(+(-\)\(57\!\cdots\!52\)\( - \)\(38\!\cdots\!84\)\( \beta_{1} + \)\(10\!\cdots\!76\)\( \beta_{2} + \)\(48\!\cdots\!48\)\( \beta_{3}) q^{84}\) \(+(-\)\(22\!\cdots\!40\)\( + \)\(16\!\cdots\!50\)\( \beta_{1} - \)\(24\!\cdots\!94\)\( \beta_{2} - \)\(69\!\cdots\!00\)\( \beta_{3}) q^{85}\) \(+(\)\(34\!\cdots\!04\)\( - \)\(62\!\cdots\!04\)\( \beta_{1} - \)\(45\!\cdots\!32\)\( \beta_{2} + \)\(28\!\cdots\!64\)\( \beta_{3}) q^{86}\) \(+(\)\(50\!\cdots\!80\)\( - \)\(45\!\cdots\!46\)\( \beta_{1} + \)\(41\!\cdots\!10\)\( \beta_{2} + \)\(11\!\cdots\!55\)\( \beta_{3}) q^{87}\) \(+(-\)\(50\!\cdots\!44\)\( - \)\(41\!\cdots\!60\)\( \beta_{1} + \)\(76\!\cdots\!12\)\( \beta_{2} - \)\(73\!\cdots\!24\)\( \beta_{3}) q^{88}\) \(+(-\)\(27\!\cdots\!90\)\( + \)\(39\!\cdots\!30\)\( \beta_{1} + \)\(13\!\cdots\!50\)\( \beta_{2} - \)\(57\!\cdots\!00\)\( \beta_{3}) q^{89}\) \(+(\)\(30\!\cdots\!40\)\( + \)\(12\!\cdots\!00\)\( \beta_{1} - \)\(19\!\cdots\!56\)\( \beta_{2} - \)\(10\!\cdots\!00\)\( \beta_{3}) q^{90}\) \(+(\)\(19\!\cdots\!72\)\( - \)\(94\!\cdots\!00\)\( \beta_{1} - \)\(38\!\cdots\!88\)\( \beta_{2} + \)\(24\!\cdots\!76\)\( \beta_{3}) q^{91}\) \(+(\)\(51\!\cdots\!48\)\( + \)\(63\!\cdots\!80\)\( \beta_{1} + \)\(33\!\cdots\!36\)\( \beta_{2} + \)\(14\!\cdots\!28\)\( \beta_{3}) q^{92}\) \(+(\)\(30\!\cdots\!44\)\( - \)\(68\!\cdots\!52\)\( \beta_{1} + \)\(51\!\cdots\!16\)\( \beta_{2} - \)\(11\!\cdots\!32\)\( \beta_{3}) q^{93}\) \(+(\)\(25\!\cdots\!72\)\( + \)\(45\!\cdots\!00\)\( \beta_{1} + \)\(10\!\cdots\!24\)\( \beta_{2} + \)\(14\!\cdots\!52\)\( \beta_{3}) q^{94}\) \(+(\)\(90\!\cdots\!00\)\( - \)\(12\!\cdots\!50\)\( \beta_{1} - \)\(17\!\cdots\!90\)\( \beta_{2} + \)\(10\!\cdots\!75\)\( \beta_{3}) q^{95}\) \(+(-\)\(53\!\cdots\!76\)\( + \)\(52\!\cdots\!68\)\( \beta_{1}) q^{96}\) \(+(\)\(21\!\cdots\!46\)\( + \)\(30\!\cdots\!58\)\( \beta_{1} - \)\(46\!\cdots\!78\)\( \beta_{2} - \)\(25\!\cdots\!44\)\( \beta_{3}) q^{97}\) \(+(-\)\(10\!\cdots\!64\)\( + \)\(62\!\cdots\!32\)\( \beta_{1} + \)\(14\!\cdots\!96\)\( \beta_{2} - \)\(16\!\cdots\!92\)\( \beta_{3}) q^{98}\) \(+(\)\(55\!\cdots\!44\)\( + \)\(37\!\cdots\!33\)\( \beta_{1} + \)\(18\!\cdots\!12\)\( \beta_{2} + \)\(28\!\cdots\!76\)\( \beta_{3}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 35184372088832q^{2} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!28\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!56\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!60\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!24\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(85\!\cdots\!04\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!48\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(82\!\cdots\!48\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 35184372088832q^{2} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!28\)\(q^{3} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!56\)\(q^{4} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!60\)\(q^{5} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!24\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(85\!\cdots\!04\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!48\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(82\!\cdots\!48\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!80\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!48\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!92\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!12\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(75\!\cdots\!32\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!80\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!84\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(70\!\cdots\!16\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(72\!\cdots\!84\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!60\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!40\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!72\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!84\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!28\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!36\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!00\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!96\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(98\!\cdots\!00\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(66\!\cdots\!56\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(93\!\cdots\!40\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!40\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!68\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!72\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(97\!\cdots\!64\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!28\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!40\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(63\!\cdots\!72\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!64\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!80\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!84\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!20\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!08\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!76\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!52\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!72\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!20\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!24\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!36\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!88\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!32\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(91\!\cdots\!12\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!68\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!12\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(86\!\cdots\!00\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!20\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!48\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!20\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(82\!\cdots\!20\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(81\!\cdots\!80\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!20\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(90\!\cdots\!28\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!44\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!48\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!76\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!80\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(85\!\cdots\!12\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(65\!\cdots\!96\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!24\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!04\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!20\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!92\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!76\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!88\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!12\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!40\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!48\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!72\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!20\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!60\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!76\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(48\!\cdots\!64\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!68\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!08\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(89\!\cdots\!60\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!16\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!20\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!76\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!60\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!60\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!88\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!92\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!76\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!88\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!04\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(84\!\cdots\!84\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!56\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!76\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(2\) \(x^{3}\mathstrut -\mathstrut \) \(281103089248735483936525960163617359\) \(x^{2}\mathstrut -\mathstrut \) \(12802046688892212650602531506830185803155982810145588\) \(x\mathstrut +\mathstrut \) \(13431538901013885043865389957038837902382364731827315409437340098095268\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 1920 \nu - 960 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(9433750962176\) \(\nu^{3}\mathstrut +\mathstrut \) \(1769549267668589556685668835328\) \(\nu^{2}\mathstrut +\mathstrut \) \(1629454524675798842203795893047621524173694499456\) \(\nu\mathstrut -\mathstrut \) \(158134392657869780369087954106442708349250693952206168207850810688\)\()/\)\(52\!\cdots\!57\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(10114407221690368\) \(\nu^{3}\mathstrut +\mathstrut \) \(12622111923347229353644701057531904\) \(\nu^{2}\mathstrut +\mathstrut \) \(1014369494928421110790943986822141502075067313900288\) \(\nu\mathstrut -\mathstrut \) \(1676943492136190755850952090281228658060462045559125237479226421642624\)\()/\)\(57\!\cdots\!27\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(960\)\()/1920\)
\(\nu^{2}\)\(=\)\((\)\(98963986\) \(\beta_{3}\mathstrut -\mathstrut \) \(9645850393\) \(\beta_{2}\mathstrut +\mathstrut \) \(65580735096668436823\) \(\beta_{1}\mathstrut +\mathstrut \) \(259064607051634621995902324886789759897600\)\()/1843200\)
\(\nu^{3}\)\(=\)\((\)\(2376103752972959609807416649\) \(\beta_{3}\mathstrut -\mathstrut \) \(1540785146892972333208581672062\) \(\beta_{2}\mathstrut +\mathstrut \) \(22799155987924903377416244937247733974012\) \(\beta_{1}\mathstrut +\mathstrut \) \(2265286315868748229809505370873077099774695078030660953702400\)\()/\)\(235929600\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
5.03541e17
2.13103e17
−3.01536e17
−4.15109e17
−8.79609e12 −8.65990e20 7.73713e25 −4.55634e30 7.61733e33 8.09122e36 −6.80565e38 4.26681e41 4.00780e43
1.2 −8.79609e12 −3.08348e20 7.73713e25 1.64822e30 2.71226e33 −1.87890e36 −6.80565e38 −2.28179e41 −1.44979e43
1.3 −8.79609e12 6.79758e20 7.73713e25 −4.34651e30 −5.97921e33 −6.14091e36 −6.80565e38 1.38813e41 3.82323e43
1.4 −8.79609e12 8.97819e20 7.73713e25 3.21672e30 −7.89730e33 8.51076e36 −6.80565e38 4.82821e41 −2.82946e43
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{4} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!28\)\( T_{3}^{3} \) \(\mathstrut -\mathstrut \)\(97\!\cdots\!56\)\( T_{3}^{2} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!28\)\( T_{3} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!76\)\( \) acting on \(S_{88}^{\mathrm{new}}(\Gamma_0(2))\).